Abstract
In order to progress from the relatively harmless avascular state to the potentially lethal vascular state, solid tumours must induce the growth of new blood vessels from existing ones, a process called angiogenesis. The capillary growth centres around endothelial cells: there are several cell-based models of this process in the literature and these have reproduced some of the key microscopic features of capillary growth. The most common approach is to simulate the movement of leading endothelial cells on a regular lattice. Here, we apply a circular random walk model to the process of angiogenesis, and thus allow the cells to move independently of a lattice; the results display good agreement with empirical observations. We also run simulations of two lattice-based models in order to make a critical comparison of the different modelling approaches. Finally, non-lattice simulations are carried out in the context of a realistic model of tumour angiogenesis, and potential anti-angiogenic strategies are evaluated.
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References
Anderson, A. R. A. and M. A. J. Chaplain (1998). Continuous and discrete mathematical models of tumour-induced angiogenesis. Bull. Math. Biol. 60, 857–900.
Anderson, A. R. A., M. A. J. Chaplain, E. L. Newman, R. J. C. Steele and A. M. Thompson (2000). Mathematical modelling of tumour invasion and metastasis. J. Theor. Med. 2, 129–154.
Anderson, A. R. A., B. D. Sleeman, I. M. Young and B. S. Griffiths (1997). Nematode movement along a chemical gradient in a structurally heterogeneous environment. 2. Theory. Fundan. Appl. Nematol. 20, 165–172.
Ausprunk, D. H. and J. Folkman (1977). Migration and proliferation of endothelial cells in preformed and newly formed blood vessels during tumour angiogenesis. Microvasc. Res. 14, 53–65.
Balding, D. and D. L. S. McElwain (1985). Mathematical modelling of tumour-induced capillary growth. J. Theor. Biol. 114, 53–73.
Bowersox, J. C. and N. Sorgente (1982). Chemotaxis of aortic endothelial cells in response to fibronectin. Cancer Res. 42, 2547–2551.
Brem, H. and J. Folkman (1975). Inhibition of tumour angiogenesis mediated by cartilage. J. Exp. Med. 141, 427–439.
Burn, D. S., (2000). Gene flow in agricultural systems and angiogenesis in tumour growth. PhD thesis, School of Mathematics, University of Leeds.
Carmeliet, P. and R. K. Jain (2000). Angiogenesis in cancer and other diseases. Nature 407, 249–257.
Carter, S. B. (1965). Principles of cell motility: the direction of cell movement and cancer invasion. Nature 208, 1183–1187.
Chaplain, M. A. J., S. M. Giles, B. D. Sleeman and R. J. Jarvis (1995). A mathematical model for tumour angiogenesis. J. Math. Biol. 33, 744–770.
Chaplain, M. A. J. and A. M. Stuart (1993). A model mechanism for the chemotactic response of endothelial cells to tumour angiogenesis factor. IMA J. Math. Appl. Med. Biol. 10, 149–168.
Clark, R. A. F., H. F. Dvorak and R. B. Colvin (1981). Fibronectin in delayed-type hypersensitivity skin reactions: associations with vessel permeability and endothelial cell activation. J. Immunol. 126, 787–793.
Dallon, J. C. and H. G. Othmer (1997). A discrete cell model with adaptive signalling for aggregation of Dictyostelium discoideum. Phil. Trans. R. Soc. Lond. B 352, 391–417.
Davis, B. (1990). Reinforced random walk. Prob. Theor. Rel. Fields 84, 203–229.
Denekamp, J. and B. Hobson (1982). Endothelial cell proliferation in experimental tumours. Br. J. Cancer 46, 711–720.
Folkman, J. (1971). Tumour angiogenesis: therapeutic implications. New Engl. J. Med. 285, 1182–1186.
Folkman, J. (1974). Tumour angiogenesis. Adv. Cancer Res. 19, 331–358.
Folkman, J. and M. Klagsbrun (1987). Angiogenic factors. Science 235, 442–447.
Han, Z. C. and Y. Liu (1999). Angiogenesis: state of the art. Int. J. Haematol. 70, 68–82.
Hanahan, D. and J. Folkman (1996). Patterns and emerging mechanisms of the angiogenic switch during tumourigenesis. Cell 86, 353–364.
Hashizume, H., P. Baluk, S. Morikawa, J. W. McLean, G. Thurston, S. Roberge, R. K. Jain and D. M. McDonald (2000). Openings between defective endothelial cells explain tumour vessel leakiness. Am. J. Path. 156, 1363–1380.
Hill, N. A. and D. P. Häder (1997). A biased random walk model for the trajectories of swimming micro-organisms. J. Theor. Biol. 186, 503–526.
Hunt, T. K., D. R. Knighton, K. K. Thakral, W. H. Goodson and W. S. Andrews (1984). Studies on inflammation and wound healing: angiogenesis and collagen synthesis stimulated in vivo by resident and activated macrophages. Surgery 96, 48–54.
Klagsbrun, M. and P. A. D’Amore (1996). Vascular endothelial growth factor and its receptors. Cytokine Growth Factor Rev. 7, 259–270.
Konerding, M. A., W. Malkusch, B. Klapthor, C. van Ackern, E. Fait, S. A. Hill, C. Parkins, D. J. Chaplin, M. Presta and J. Denekamp (1999). Evidence for characteristic vascular patterns in solid tumours: quantitative studies using corrosion casts. Br. J. Cancer 80, 724–732.
Konerding, M. A., C. van Ackern, F. Steinberg and C. Streffer (1992). Combined morphological approaches in the study of network formation in tumour angiogenesis, in Angiogenesis: Key Principles—Science— Technology—Medicine, R. Steiner, P. B. Weisz and R. Langer (Eds), Basel: Birkhauser, pp. 40–58.
Less, J. R., T. C. Skalak, E. M. Sevick and R. K. Jain (1991). Microvascular architecture in a mammary carcinoma: branching patterns and vessel dimensions. Cancer Res. 51, 265–273.
Leung, D. W., G. Cachianes, W. J. Kuang, D. V. Goeddel and N. Ferrara (1989). Vascular endothelial growth factor is a secreted angiogenic mitogen. Science 246, 1306–1309.
Levine, H. A., S. Pamuk, B. D. Sleeman and M. Nilsen-Hamilton (2001a). A mathematical model of capillary formation and development in tumour angiogenesis: penetration into the stroma. Bull. Math. Biol. 63, 801–863.
Levine, H. A. and B. D. Sleeman (1997). A system of reaction diffusion equations arising in the theory of reinforced random walks. SIAM J. Appl. Math. 57, 683–730.
Levine, H. A., B. D. Sleeman and M. Nilsen-Hamilton (2001b). Mathematical modelling of the onset of capillary formation initiating angiogenesis. J. Math. Biol. 42, 195–238.
Liotta, L. A., P. S. Steeg and W. G. Stetler-Stevenson (1991). Cancer metastasis and angiogenesis: an imbalance of positive and negative regulation. Cell 64, 327–336.
Moses, M. A. and R. Langer (1992). Metalloproteinase inhibition as a mechanism for the inhibition of angiogenesis, in Angiogenesis: Key Principles—Science—Technology—Medicine, R. Steiner, P. B. Weisz and R. Langer (Eds), Basel: Birkhauser, pp. 146–151.
Muthukkaruppan, V. R. and R. Auerbach (1979). Angiogenesis in the mouse cornea. Science 205, 1416–1418.
Muthukkaruppan, V. R., L. Kubai and R. Auerbach (1982). Tumour-induced neovascularisation in the mouse eye. J. Natl. Cancer Inst. 69, 699–708.
Nicosia, R. F., E. Bonanno and M. Smith (1993). Fibronectin promotes the elongation of microvessels during angiogenesis in vitro. J. Cell Physiol. 154, 654–661.
Othmer, H. G. and A. Stevens (1997). Aggregation, blowup and collapse: the ABC’s of taxis and reinforced random walks. SIAM J. Appl. Math. 57, 1044–1081.
Paweletz, N. and M. Kneirim (1989). Tumour-related angiogenesis. Crit. Rev. Oncol. Haematol. 9, 197–242.
Pepper, M. S. (1997). Manipulating angiogenesis. Arterioscter Thromb. Vasc. Biol. 17, 605–619.
Pepper, M. S. (2001). Extracellular proteolysis and angiogenesis. Thromb. Haemost. 86, 346–355.
Pepper, M. S., D. Belin, R. Montesano, L. Orci and J. D. Vassalli (1990). Transforming growth factor-β1 modulates basic fibroblast growth factor-induced proteolytic and angiogenic properties of endothelial cells in vitro. J. Cell Biol. 111, 743–755.
Plank, M. J. and B. D. Sleeman (2003). A reinforced random walk model of tumour angiogenesis and anti-angiogenic strategies. IMA J. Math. Med. Biol. 20, 135–181.
Plank, M. J., B. D. Sleeman and P. F. Jones (2003). A mathematical model of an in vitro experiment to investigate endothelial cell migration. J. Theor. Med. 4, 251–270.
Reynolds, L. P., S. D. Killilea and D. A. Redmer (1992). Angiogenesis in the female reproductive cycle. FASEB J. 6, 886–892.
Risau, V. (1997). Mechanisms of angiogenesis. Nature 386, 671–674.
Schirrmacher, V. (1985). Cancer metastasis: experimental approaches, theoretical concepts and impacts for treatment strategies. Adv. Cancer Res. 43, 1–73.
Sholley, M. M., G. P. Ferguson, H. R. Seibel, J. L. Montour and J. D. Wilson (1984). Mechanisms of neovascularisation. Lab. Invest. 51, 624–634.
Shweiki, D., A. Itin, D. Soffer and E. Keshet (1992). Vascular endothelial growth factor induced by hypoxia may mediate hypoxia-initiated angiogenesis. Nature 359, 843–845.
Sleeman, B. D. and I. P. Wallis (2002). Tumour induced angiogenesis as a reinforced random walk: modelling capillary network formation without endothelial cell proliferation. J. Math. Comp. Modelling 36, 339–358.
Stokes, C. L. and D. A. Lauffenburger (1991). Analysis of the roles of microvessel endothelial cell random motility and chemotaxis in angiogenesis. J. Theor. Biol. 152, 377–403.
Unemori, E. N., N. Ferrara, E. A. Bauer and E. P. Amento (1992). Vascular endothelial growth factor induces interstitial collagenase expression in human endothelial cells. J. Cell Physiol. 153, 557–562.
Vajkoczy, P., M. Farhadi, A. Gaumann, R. Heidenreich, R. Erber, A. Wunder, J. C. Tonn, M. D. Menger and G. Breier (2002). Microtumour growth initiates angiogenic sprouting with simultaneous expression of VEGF, VEGF receptor-2 and angiopoietin-2. J. Clin. Invest. 109, 777–785.
Vernon, R. B. and E. H. Sage (1999). A novel, quantitative model for study of endothelial cell migration and sprout formation within three-dimensional collagen matrices. Microvasc. Res. 57, 118–133.
Yancopoulos, G. D., S. Davis, N. W. Gale, J. S. Rudge, S. J. Wiegand and J. Holash (2000). Vascular-specific growth factors and blood vessel formation. Nature 407, 242–249.
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Plank, M.J., Sleeman, B.D. Lattice and non-lattice models of tumour angiogenesis. Bull. Math. Biol. 66, 1785–1819 (2004). https://doi.org/10.1016/j.bulm.2004.04.001
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DOI: https://doi.org/10.1016/j.bulm.2004.04.001